Smooth conditional measures for strong stable foliations of Anosov flows

In general, for the geodesic flow of a compact negatively curved manifold for example, the strong stable foliation is Holder continuous; in particular, even if each leaf is (locally) a smooth manifold, transversally, the foliation (and therefore the map $x\to\delta_x$) is no more than Hölder continuous.

In the case of geodesic flows in constant negative curvature, it is smooth, and if I remember well, it is $C^1$ in the case of surfaces.

I guess that for general Anosov flows, it is similar: you cannot expect better regularity than Hölder, except in very particular cases.